The GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) is a radio continuum survey at 76-227MHz of the entire southern sky (Declination <+30{deg}) with an angular resolution of ~2arcmin. In this paper, we combine GLEAM data with optical spectroscopy from the 6dF Galaxy Survey to construct a sample of 1,590 local (median z~0.064) radio sources with S_200MHz_>55mJy across an area of ~16700deg^2^. From the optical spectra, we identify the dominant physical process responsible for the radio emission from each galaxy: 73 per cent are fuelled by an active galactic nucleus (AGN) and 27 per cent by star formation. We present the local radio luminosity function for AGN and star-forming galaxies at 200MHz and characterise the typical radio spectra of these two populations between 76MHz and ~1GHz. For the AGN, the median spectral index between 200MHz and ~1GHz, {alpha}, is -0.600+/-0.010 (where S proportional to {nu}^{alpha}^) and the median spectral index within the GLEAM band, {alpha}_low_, is -0.704+/-0.011. For the star-forming galaxies, the median value of {alpha}_high_ is -0.650+/-0.010 and the median value of {alpha}_low_ is -0.596+/-0.015. Among the AGN population, flat-spectrum sources are more common at lower radio luminosity, suggesting the existence of a significant population of weak radio AGN that remain core-dominated even at low frequencies. However, around 4 per cent of local radio AGN have ultra-steep radio spectra at low frequencies ({alpha}_low_<-1.2). These ultra-steep-spectrum sources span a wide range in radio luminosity, and further work is needed to clarify their nature.